Editing Inductive reasoning (section)

==Inductive inference==
Around 1960, [[Ray Solomonoff]] founded the theory of universal [[Solomonoff's theory of inductive inference|inductive inference]], a theory of prediction based on observations, for example, predicting the next symbol based upon a given series of symbols. This is a formal inductive framework that combines [[algorithmic information theory]] with the Bayesian framework. Universal inductive inference is based on solid philosophical foundations and 'seems to be an inadequate tool for dealing with any reasonably complex or real-world environment',<ref>{{cite journal |first1=Samuel |last1=Rathmanner |author2-link=Marcus Hutter |first2=Marcus |last2=Hutter |title=A Philosophical Treatise of Universal Induction |journal=Entropy |volume=13 |issue=6 |pages=1076–136 |year=2011 |doi=10.3390/e13061076 |bibcode=2011Entrp..13.1076R |arxiv=1105.5721 |s2cid=2499910 |doi-access=free }}</ref> and can be considered as a mathematically formalized [[Occam's razor]]. Fundamental ingredients of the theory are the concepts of [[algorithmic probability]] and [[Kolmogorov complexity]].

Inductive inference typically considers hypothesis classes with a countable size. A recent advance<ref>{{cite journal |last=Lu |first=Z. |title=When is Inductive Inference Possible? |journal=NeurIPS 2024 |year=2024 |url=https://openreview.net/pdf?id=2aGcshccuV}}</ref> established a sufficient and necessary condition for inductive inference: a finite error bound is guaranteed if and only if the hypothesis class is a countable union of online learnable classes. Notably, this condition allows the hypothesis class to have an uncountable size while remaining learnable within this framework.